corrected some pages

Several small mistakes/errors were corrected in several proofs/definitions.

Author: JoramSoch

D/chi2.md

@@ -45,7 +45,7 @@ $$\label{eq:chi2}
 Y = \sum_{i=1}^{k} X_{i}^{2} \sim \chi^{2}(k) \quad \text{where} \quad k > 0 \; .
 $$
 
-A [random variable](/D/rvar) $Y$ is said said to follow a chi-square distribution with $k$ number of degress of freedom, if and only if its [probability density function](/D/pdf) is given by
+The [probability density function of the chi-square distribution](/P/chi2-pdf) with $k$ degress of freedom is
 
 $$ \label{eq:chi2-pdf}
 \chi^{2}(x; k) = \frac{1}{2^{k/2}\Gamma (k/2)} \, x^{k/2-1} \, e^{-x/2}

D/eblme.md

@@ -39,7 +39,7 @@ $$ \label{eq:ML}
 p(y \vert \lambda, m) = \int p(y \vert \theta, \lambda, m) \, (\theta \vert \lambda, m) \, \mathrm{d}\theta
 $$
 
-and
+[and](/D/prior-eb)
 
 $$ \label{eq:EB}
 \hat{\lambda} = \operatorname*{arg\,max}_{\lambda} \log p(y \vert \lambda, m) \; .

D/prior-emp.md

@@ -28,7 +28,7 @@ username: "JoramSoch"
 ---
 
 
-**Definition:** Let $p(\theta \vert m)$ a [prior distribution](/D/prior) for the parameter $\theta$ of a [generative model](/D/gm) $m$. Then,
+**Definition:** Let $p(\theta \vert m)$ be a [prior distribution](/D/prior) for the parameter $\theta$ of a [generative model](/D/gm) $m$. Then,
 
 * the distribution is called an "empirical prior", if it has been [derived from empirical data](/P/post-jl);
 

D/prior-flat.md

@@ -27,15 +27,15 @@ sources:
     in: "NeuroImage"
     pages: "vol. 16, iss. 2, pp. 484-512, fn. 10"
     url: "https://www.sciencedirect.com/science/article/pii/S1053811902910918"
-    doi: "10.1006/nimg.2002.1091"^
+    doi: "10.1006/nimg.2002.1091"
 
 def_id: "D116"
 shortcut: "prior-flat"
 username: "JoramSoch"
 ---
 
 
-**Definition:** Let $p(\theta \vert m)$ a [prior distribution](/D/prior) for the parameter $\theta$ of a [generative model](/D/gm) $m$. Then,
+**Definition:** Let $p(\theta \vert m)$ be a [prior distribution](/D/prior) for the parameter $\theta$ of a [generative model](/D/gm) $m$. Then,
 
 * the distribution is called a "flat prior", if its [precision](/D/prec) is zero or [variance](/D/var) is infinite;
 

D/prior-inf.md

@@ -28,10 +28,10 @@ username: "JoramSoch"
 ---
 
 
-**Definition:** Let $p(\theta \vert m)$ a [prior distribution](/D/prior) for the parameter $\theta$ of a [generative model](/D/gm) $m$. Then,
+**Definition:** Let $p(\theta \vert m)$ be a [prior distribution](/D/prior) for the parameter $\theta$ of a [generative model](/D/gm) $m$. Then,
 
 * the distribution is called an "informative prior", if it biases the parameter towards particular values;
 
 * the distribution is called a "weakly informative prior", if it mildly [influences the posterior distribution](/P/post-jl);
 
-* the distribution is called a "non-informative prior", if it does not influence the [posterior hyperparameters](/D/post).
+* the distribution is called a "non-informative prior", if it does not [influence](/P/post-jl) the [posterior hyperparameters](/D/post).

D/prior-ref.md

@@ -30,5 +30,5 @@ username: "JoramSoch"
 **Definition:** Let $m$ be a [generative model](/D/gm) with [likelihood function](/D/lf) $p(y \vert \theta, m)$ and [prior distribution](/D/prior) $p(\theta \vert \lambda, m)$ using [prior hyperparameters](/D/prior) $\lambda$. Let $p(\theta \vert y, \lambda, m)$ be the [posterior distribution](/D/post) that is [proportional to the the joint likelihood](/P/post-jl). Then, the prior distribution is called a "reference prior", if it maximizes the [expected](/D/mean) [Kullback-Leibler divergence](/D/kl) of the posterior distribution relative to the prior distribution:
 
 $$ \label{eq:prior-ref}
-\lambda_{\mathrm{ref}} = \operatorname*{arg\,max}_{\lambda} \mathrm{KL} \left[ p(\theta \vert y, \lambda, m) \, || \, p(\theta \vert \lambda, m) \right] \; .
+\lambda_{\mathrm{ref}} = \operatorname*{arg\,max}_{\lambda} \left\langle \mathrm{KL} \left[ p(\theta \vert y, \lambda, m) \, || \, p(\theta \vert \lambda, m) \right] \right\rangle \; .
 $$

D/prior-uni.md

@@ -27,8 +27,8 @@ username: "JoramSoch"
 ---
 
 
-**Definition:** Let $p(\theta \vert m)$ a [prior distribution](/D/prior) for the parameter $\theta \in \Theta$ of a [generative model](/D/gm) $m$. Then,
+**Definition:** Let $p(\theta \vert m)$ be a [prior distribution](/D/prior) for the parameter $\theta$ of a [generative model](/D/gm) $m$ where $\theta$ belongs to the parameter space $\Theta$. Then,
 
-* the distribution is called a "uniform prior", if its [density](/D/pdf) is constant over the entire parameter space $\Theta$;
+* the distribution is called a "uniform prior", if its [density](/D/pdf) or [mass](/D/pmf) is constant over $\Theta$;
 
-* the distribution is called a "non-uniform prior", if its [density](/D/pdf) is not constant over the parameter space $\Theta$.
+* the distribution is called a "non-uniform prior", if its [density](/D/pdf) or [mass](/D/pmf) is not constant over $\Theta$.

D/rvar-uni.md

@@ -29,7 +29,7 @@ username: "JoramSoch"
 
 **Definition:** Let $X$ be a [random variable](/D/rvar) with possible outcomes $\mathcal{X}$. Then,
 
-* $X$ is called a two-valued random variable or [random event](/D/reve), if $\mathcal{X}$ has exactly two elements, e.g. $\mathcal{X} = \left\lbrace \mathrm{true}, \mathrm{false} \right\rbrace$ or $\mathcal{X} = \left\lbrace 1, 0 \right\rbrace$;
+* $X$ is called a two-valued random variable or [random event](/D/reve), if $\mathcal{X}$ has exactly two elements, e.g. $\mathcal{X} = \left\lbrace E, \overline{E} \right\rbrace$ or $\mathcal{X} = \left\lbrace \mathrm{true}, \mathrm{false} \right\rbrace$ or $\mathcal{X} = \left\lbrace 1, 0 \right\rbrace$;
 
 * $X$ is called a univariate random variable or [random scalar](/D/rvar), if $\mathcal{X}$ is one-dimensional, i.e. (a subset of) the real numbers $\mathbb{R}$;
 

P/dent-add.md

@@ -51,36 +51,36 @@ $$
 If $a > 0$, then $g(X)$ is a [strictly increasing function, such that the probability density function](/P/pdf-sifct) of $Y$ is
 
 $$ \label{eq:Y-pdf-c1}
-f_Y(y) = f_X(g^{-1}(y)) \, \frac{\mathrm{d}g^{-1}(y)}{\mathrm{d}y} \overset{\eqref{eq:X-Y}}{=} \frac{1}{a} \, f_X\left(\frac{y}{a}\right) \; ,
+f_Y(y) = f_X(g^{-1}(y)) \, \frac{\mathrm{d}g^{-1}(y)}{\mathrm{d}y} \overset{\eqref{eq:X-Y}}{=} \frac{1}{a} \, f_X\left(\frac{y}{a}\right) \; ;
 $$
 
-and if $a < 0$, then $g(X)$ is a [strictly decreasing function, such that the probability density function](/P/pdf-sifct) of $Y$ is
+if $a < 0$, then $g(X)$ is a [strictly decreasing function, such that the probability density function](/P/pdf-sdfct) of $Y$ is
 
 $$ \label{eq:Y-pdf-c2}
-f_Y(y) = - f_X(g^{-1}(y)) \, \frac{\mathrm{d}g^{-1}(y)}{\mathrm{d}y} \overset{\eqref{eq:X-Y}}{=} -\frac{1}{a} \, f_X\left(\frac{y}{a}\right) \; ,
+f_Y(y) = - f_X(g^{-1}(y)) \, \frac{\mathrm{d}g^{-1}(y)}{\mathrm{d}y} \overset{\eqref{eq:X-Y}}{=} -\frac{1}{a} \, f_X\left(\frac{y}{a}\right) \; ;
 $$
 
-such that we can write
+thus, we can write
 
 $$ \label{eq:Y-pdf}
-f_Y(y) = \left| \frac{1}{a} \right| \, f_X\left(\frac{y}{a}\right) \; .
+f_Y(y) = \frac{1}{|a|} \, f_X\left(\frac{y}{a}\right) \; .
 $$
 
 Writing down the differential entropy for $Y$, we have:
 
 $$ \label{eq:Y-dent-s1}
 \begin{split}
 \mathrm{h}(Y) &= - \int_{\mathcal{Y}} f_Y(y) \log f_Y(y) \, \mathrm{d}y \\
-&\overset{\eqref{eq:Y-pdf}}{=} - \int_{\mathcal{Y}} \left| \frac{1}{a} \right| \, f_X\left(\frac{y}{a}\right) \log \left[ \left| \frac{1}{a} \right| \, f_X\left(\frac{y}{a}\right) \right] \, \mathrm{d}y
+&\overset{\eqref{eq:Y-pdf}}{=} - \int_{\mathcal{Y}} \frac{1}{|a|} \, f_X\left(\frac{y}{a}\right) \log \left[ \frac{1}{|a|} \, f_X\left(\frac{y}{a}\right) \right] \, \mathrm{d}y
 \end{split}
 $$
 
 Substituting $x = y/a$, such that $y = ax$, this yields:
 
 $$ \label{eq:Y-dent-s2}
 \begin{split}
-\mathrm{h}(Y) &= - \int_{\left\lbrace y/a \,|\, y \in {\mathcal{Y}} \right\rbrace} \left| \frac{1}{a} \right| \, f_X\left(\frac{ax}{a}\right) \log \left[ \left| \frac{1}{a} \right| \, f_X\left(\frac{ax}{a}\right) \right] \, \mathrm{d}(ax) \\
-&= - \int_{\mathcal{X}} f_X(x) \log \left[ \left| \frac{1}{a} \right| \, f_X(x) \right] \, \mathrm{d}x \\
+\mathrm{h}(Y) &= - \int_{\left\lbrace y/a \,|\, y \in {\mathcal{Y}} \right\rbrace} \frac{1}{|a|} \, f_X\left(\frac{ax}{a}\right) \log \left[ \frac{1}{|a|} \, f_X\left(\frac{ax}{a}\right) \right] \, \mathrm{d}(ax) \\
+&= - \int_{\mathcal{X}} f_X(x) \log \left[ \frac{1}{|a|} \, f_X(x) \right] \, \mathrm{d}x \\
 &= - \int_{\mathcal{X}} f_X(x) \left[ \log f_X(x) - \log |a| \right] \, \mathrm{d}x \\
 &= - \int_{\mathcal{X}} f_X(x) \log f_X(x) \, \mathrm{d}x + \log |a| \int_{\mathcal{X}} f_X(x) \, \mathrm{d}x \\
 &\overset{\eqref{eq:X-dent}}{=} \mathrm{h}(X) + \log |a| \; .

P/dent-inv.md

@@ -45,7 +45,7 @@ where $p(x) = f_X(x)$ is the [probability density function](/D/pdf) of $X$.
 Define the mappings between $X$ and $Y = X + c$ as
 
 $$ \label{eq:X-Y}
-Y = g(X) = X + c \quad \Leftrightarrow \quad X = g^{-1}(Y) = X - c \; .
+Y = g(X) = X + c \quad \Leftrightarrow \quad X = g^{-1}(Y) = Y - c \; .
 $$
 
 Note that $g(X)$ is a [strictly increasing function, such that the probability density function](/P/pdf-sifct) of $Y$ is